eli5: What is multimodal distribution? With examples, please 🙂
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Consider height. Adult women are about five inches shorter than adult men. If you look at just the distribution of women’s height, it’s a normal distribution – many clustered around the mean, few at the extremes. Likewise, if you look at just the distribution of men’s height, it’s a normal distribution.
But if you look at the distribution of all human height, you end up with a bimodal distribution. The mode (most common value) of female height is about five inches less than the mode of male height while the variance is roughly the same. Adding them together preserves these two modes, thus ‘bimodal’ or ‘multimodal’.
In general, when you see multimodal distributions in real world data, it implies that you’re combining two or more distributions that have different characteristics.
English answer. You buy something from China. Its packed into a container and taken by lorry to the port where its transferred to a ship and then brought to your country where it is unloaded, transferred to a rail car and shipped to your area. It is then put on a ruck for the final part of the journey to your site. I guess I’ve worked in Logistics too long.
A multimodal distribution is a continuous probability distribution with two or more modes. Like two or more local maxima in the probability density function. Modes are points in the distribution where the probability density is highest, and a distribution can have more than one mode if it has more than one peak.
It can be bimodal distributions, which are often used to describe data sets that have two distinct groups or clusters, such as the heights of men and women in a population. There is trimodal… Etc.
There more mode the more diffucult, there are specialized statistical techniques and tools available for analyzing multimodal data, such as nonparametric density estimation methods and mixture models but it’s not Eli5 anymore (mmmh it’s never been I think).
I don’t think there is a clear unambiguous definition that captures all cases (correct me on that if there is one), but a multimodal distribution is one where there are multiple “humps”, basically. “Bimodal” means there are exactly two.
For example, the distribution of “time it takes for you to trim a nail” – your fingernails grow something like 4x faster than toenails, so there will be a lot of probability mass near one value, and a lot of it near some other value 4x larger, with not a lot inbetween. You can’t really simplify that by replacing it with a single gaussian. Or the distribution of Euler angles of a flipped coin – the “azimuth” will be uniformly distributed, but the other angle will be either 0 or 180 degrees, for heads/tails. Or the distribution of speeds in a multi-lane highway – depending on how traffic flows, there could be one “mode” per lane, or one for cars and one for large trucks, or they all mix together. Or sometimes the distribution of pixel colors in an image – large similarly shaded areas will correspond to peaks in a histogram, and for many images there will be multiple distinct ones. Or the distribution of flow rate of a sink – it’s either on or off most of the time, and the “on” amount varies, but it’s rarely left to trickle.
Mode refers to the peak (most numerous) value or range. Multimodal is just saying that there is more than one peak to the data as plotted.
The height of human adults is a good example, because there will be two peaks in the data distribution, one corresponding to the modal (most common) height for women and the other corresponding to the modal height from men.
There are lots of reasons why a distribution could display more than one peak. In the human height example, the existence of a bimodal distribution is a strong indicator of some external factor influencing the measurements (there are actually two different populations being measured and combined together, and the two populations are distinct with regard to the measured parameters, the things being measured). The existence of the bimodal distribution is declaring that something is responsible that has not been considered (distinctions by population was not expected for some reason or other, so time to figure out what that might be).
In rare cases, the multimodal (many peaks) distribution can be due to sampling bias (different sampling methods can lead to collection of items that are different simply because the sampling methods each favor one size range over another, and they are different ranges, so the results cover different ranges).
If you were to extrapolate this idea to its full extent (measuring of many populations which are not actually related yet quite different respecting the thing being measured), you would get a distribution that is basically noise, what is called noise, literally. Background results that have nothing to do with what you want to know and show no clear patterns. Numerous factors are about equal in importance and yet produce completely different things to each other, so you get a more or less random set of results that cover a wide range in values with no obvious pattern.
Example: Attorney salaries have a bimodal distribution.
The mean salary is around $100k, but few attorneys actually have salaries in that range.
About half of attorneys earn around $60k, and 25% of attorneys earn more than $200k, with few attorneys in the middle.
The graph looks like 2 humps:
[Attorney Bimodal Salary Distribution ](https://www.nalp.org/salarydistrib#2021)
Other people have explained the base concept pretty well already, but for another example, the Old Faithful geyser in Yellowstone has a bimodal (two mode) distribution of minutes between eruptions – although eruptions can happen between 35 and 120 minutes after the last one, they tend to happen at either 50ish or 80ish minutes after the last eruption.
The reason for this is directly related to the previous eruption time (which is also bimodal). Eruptions last 1.5 to 5 seconds. When they are shorter, the break between eruptions isn’t as long. When they are longer, the break is also longer. So if you know how long the blast from the last eruption lasted, you can pretty easily predict when the next eruption will happen, but if you don’t know that information, you just know that the most likely wait time will be around 50 minutes or around 80 minutes.
You familiar with a bell curve? One bump that spreads out. Multi-modal distributions have multiple bumps, usually because of different (or perhaps periodic) causes.
An example is motorcycle fatalities. If you plot out motorcycle fatalities by age, you get two bumps (a bi-modal distribution). One bump is in the low 20s and one bump is around 50. If you were to dig deeper, you could split up the data by accident type. Then you’d see a bunch of young folks with sport bikes going too fast into the sides of cars and a bunch of midlife gentlemen buying their cruisers and riding them to a bar to meet with like-minded fellas once a month only to forget how to counter-steer on the way home and turning right off the road when the road turned left.
Another (this time, completely made up) example is the lifespan of an iPhone. Number of phones broken vs time of ownership. I saw a lot of videos of people opening their iPhone only to immediately drop it on the pavement. There’s another bump right around a year when the new model comes out and spoiled little brats break their phone so that they need a new one. This annual spike repeats with some decay (fewer and fewer phones survive to the next year).
Call length in call centres.
Tons of very short calls (less than one minute).
Smaller number of calls in the middle
Tons of very long sales calls.
So, a horizontal histogram would be: (in 30 second increments)
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There are two distinct modes, one for the “hi, how are you? Can’t talk today”, and one for “Ya, we need to get that. What color does it come in?”.
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